OBRABOTKAMETALLOV technology Vol. 25 No. 1 2023 Introduction The quality of machining, as well as the cutter power, is associated with the stability of the vibration dynamics of the cutting process. Vibrations of the tool tip during cutting depend on two groups of factors, the first group is vibration activity unrelated to processes occurring in the cutting zone, and the second group is vibration activity caused by processes occurring in the cutting zone. In this regard, the development of technologies and methods for minimizing vibration activity, the nature of which is determined by the cutting process, is of great importance. The vibrations of the tool accompanying the cutting process are largely determined by the regeneration of vibrations when cutting along the “trace”, which is called the regenerative effect [1–4]. In these papers, it is noted that the main factor influencing the regenerative effect is the so-called “time delay” [5–7], it determines the stability of the process dynamics. In addition to the regenerative nature of the self-excitation vibration dynamics of the cutting system, the stability of the cutting tool vibrations is affected by: the temperature in the contact zone of the tool and the workpiece [8]; changes in the force response from the cutting process to the forming motions of the tool [9]; the value characterizing the degree of the cutting wedge wear, etc. [10]. However, the developing synergetic concept describing the processes occurring in machines [11–12] allows us to speak not about every individual factor, but about a certain set of interrelated and interacting factors that determine the mechanism of self-excitation of the cutting control system [13–14]. The most important factor in the complexity of the mathematical description of cutting processes dynamics is, already mentioned earlier, the “time delay”, which determines the regenerative nature of the self-excitation of the cutting system. It should be noted that in the process of linearization of a system of integro-differential equations describing the complex, nonlinear, delayed dynamics of the cutting process, one will have to deal with an element containing a lagging argument. Such an element will not allow an analysis of cutting control system differential equations using a linearized model in the vicinity of the equilibrium point based on algebraic criteria, such as Hurwitz criterion [15], or Raus criterion [16]. The solution to this problem is the use of frequency stability criteria, such as the Nyquist criterion [17-18], or its Soviet counterpart, the Mikhailov criterion [19-21]. The Nyquist criterion itself, applied to mathematical models of metal cutting control systems, is well considered in research papers of V.L. Zakorotny [11, 12], but Mikhailov’s criterion, well-known and long-known in the American engineering school [21], is not widely used yet. The purpose of such modeling is to determine some best cutting mode; such a mode, in which selfexcitation factor of the cutting control system will be minimized. It has already been experimentally proved that such a mode exists, and it is related to the cutting speed [22, 23]. In these research papers, the best mode is understood as the mode that provides the minimum roughness of the processed surface and the maximum dimensional stability of the cutting tool. For example, A.D. Makarov, in his monograph [22], formulates the following statement: “the most important factor determining the characteristics of the cutting process is the average contact temperature determined by the cutting mode (cutting speed).” In this and other papers, the tool contact temperature is determined by the current power released during cutting and converted into heat, which linearly depends on the cutting speed. However, in the paper [8] it was shown that when the wear of the cutting wedge along the flank is formed, an additional thermodynamic feedback is formed, which prewarms the cutting zone for the period up to the current moment of cutting. In the future, this will lead to a thermal expansion of the workpiece material, which will increase the value of the force pushing the tool. It should be noted that this factor, the restructuring of the force reaction, which is confirmed by experimental studies [9], was not previously taken into account when forming mathematical models of cutting systems. A.D. Makarov himself, in his reasoning, relied on the following well-known factors: 1) the falling nature of the cutting force (temperature-speed factor of processing), identified and presented as a graphical characteristic in the works of N.N. Zorev [24]; 2) the existence of “favorable conditions” in the cutting zone, interpreted by the transition from the adhesive nature to the diffusion nature of friction [22].
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